Asynchronous Agreement through Distributed Coordination Algorithms Associated with Periodic Matrices

For the widely studied consensus-type distributed multi-agent algorithms, a standard discrete-time model is a linear system whose system matrix is stochastic, thus implementing the “averaging” updating rule for each agent. To ensure agreement among all the agents asymptotically, one usually requires the stochastic matrix to be indecomposable and aperiodic (SIA). In this paper, we show that in practice such requirements can be relaxed by allowing the matrix to be periodic if the agents update asynchronously. Such a relaxation is somewhat surprising since for synchronous updating, periodic matrices in general give rise to oscillations in the agents’ states. The key step to prove reaching agreement is to use a stochastic Lyapunov function to prove almost sure convergence of the associated stochastic linear system. The results reveal the critical role that asynchrony may play for distributed network algorithms.